hdmap1l6i - Mathbox for Norm Megill

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Theorem hdmap1l6i 42877

Description: Lemmma for hdmap1l6 42880. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.)

**Hypotheses**
Ref	Expression
hdmap1l6.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1l6.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1l6.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1l6.p	⊢ + = (+_g‘𝑈)
hdmap1l6.s	⊢ − = (-_g‘𝑈)
hdmap1l6c.o	⊢ 0 = (0_g‘𝑈)
hdmap1l6.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1l6.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1l6.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1l6.a	⊢ + = (+_g‘𝐶)
hdmap1l6.r	⊢ 𝑅 = (-_g‘𝐶)
hdmap1l6.q	⊢ 𝑄 = (0_g‘𝐶)
hdmap1l6.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap1l6.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1l6.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1l6.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap1l6.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1l6cl.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1l6.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
hdmap1l6i.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
hdmap1l6i.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1l6i.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
hdmap1l6i.yz	⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))

**Assertion**
Ref	Expression
hdmap1l6i	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) + (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Proof of Theorem hdmap1l6i Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1l6.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1l6.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
5		hdmap1l6.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		hdmap1l6cl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
7	6	eldifad 3909	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
8		hdmap1l6i.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3909	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10	1, 2, 3, 4, 5, 7, 9	dvh3dim 42505	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
11		hdmap1l6.p	. . . 4 ⊢ + = (+_g‘𝑈)
12		hdmap1l6.s	. . . 4 ⊢ − = (-_g‘𝑈)
13		hdmap1l6c.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
14		hdmap1l6.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
15		hdmap1l6.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
16		hdmap1l6.a	. . . 4 ⊢ + = (+_g‘𝐶)
17		hdmap1l6.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
18		hdmap1l6.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
19		hdmap1l6.l	. . . 4 ⊢ 𝐿 = (LSpan‘𝐶)
20		hdmap1l6.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
21		hdmap1l6.i	. . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
22	5	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23		hdmap1l6.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
24	23	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷)
25	6	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
26		hdmap1l6.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
27	26	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
28		hdmap1l6i.xn	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
29	28	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
30		hdmap1l6i.yz	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
31	30	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
32	8	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
33		hdmap1l6i.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
34	33	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑍 ∈ (𝑉 ∖ { 0 }))
35		eqid 2761	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
36	1, 2, 5	dvhlmod 42169	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	36	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod)
38	3, 35, 4, 36, 7, 9	lspprcl 21254	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
39	38	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
40		simp2 1155	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉)
41		simp3 1156	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
42	13, 35, 37, 39, 40, 41	lssneln0 21229	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ (𝑉 ∖ { 0 }))
43	1, 2, 3, 11, 12, 13, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 27, 29, 31, 32, 34, 42, 41	hdmap1l6h 42876	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) + (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
44	43	rexlimdv3a 3168	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) + (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
45	10, 44	mpd 16	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) + (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∃wrex 3087 ∖ cdif 3896 {csn 4584 {cpr 4586 ⟨cotp 4592 ‘cfv 6539 (class class class)co 7421 Basecbs 17388 +_gcplusg 17429 0_gc0g 17611 -_gcsg 19147 LModclmod 21136 LSubSpclss 21207 LSpanclspn 21247 HLchlt 40409 LHypclh 41043 DVecHcdvh 42137 LCDualclcd 42645 mapdcmpd 42683 HDMap1chdma1 42850

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-riotaBAD 40012

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-pss 4035 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8244 df-undef 8291 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-er 8718 df-map 8850 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-n0 12608 df-z 12695 df-uz 12967 df-fz 13641 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-0g 17613 df-mre 17757 df-mrc 17758 df-acs 17760 df-proset 18469 df-poset 18488 df-plt 18503 df-lub 18519 df-glb 18520 df-join 18521 df-meet 18522 df-p0 18598 df-p1 18599 df-lat 18607 df-clat 18674 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-submnd 18980 df-grp 19148 df-minusg 19149 df-sbg 19150 df-subg 19334 df-cntz 19532 df-oppg 19561 df-lsm 19851 df-cmn 19997 df-abl 19998 df-mgp 20362 df-rng 20376 df-ur 20409 df-ring 20462 df-oppr 20568 df-dvdsr 20588 df-unit 20589 df-invr 20619 df-dvr 20632 df-nzr 20764 df-rlreg 20947 df-domn 20948 df-drng 20983 df-lmod 21138 df-lss 21208 df-lsp 21248 df-lvec 21379 df-lsatoms 40035 df-lshyp 40036 df-lcv 40078 df-lfl 40117 df-lkr 40145 df-ldual 40183 df-oposet 40235 df-ol 40237 df-oml 40238 df-covers 40325 df-ats 40326 df-atl 40357 df-cvlat 40381 df-hlat 40410 df-llines 40557 df-lplanes 40558 df-lvols 40559 df-lines 40560 df-psubsp 40562 df-pmap 40563 df-padd 40855 df-lhyp 41047 df-laut 41048 df-ldil 41163 df-ltrn 41164 df-trl 41218 df-tgrp 41802 df-tendo 41814 df-edring 41816 df-dveca 42062 df-disoa 42088 df-dvech 42138 df-dib 42198 df-dic 42232 df-dih 42288 df-doch 42407 df-djh 42454 df-lcdual 42646 df-mapd 42684 df-hdmap1 42852

This theorem is used by: hdmap1l6j 42878

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